Applying ratio in contexts and units
Ratio, proportion and rates • Ratio and proportion
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Key concepts
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Ratio in real contexts
Ratios compare two or more quantities of the same or different type and guide decisions in problems such as scaling recipes, comparing strengths or splitting amounts. Ratios require interpretation as parts of a whole, multiplicative factors or unit rates depending on context. Ratios operate only when units align or when the problem explicitly mixes different units. Incorrect unit handling leads to meaningless comparisons.
Unit conversion and consistency
Unit conversion changes a measurement into an equivalent value in a different unit while preserving quantity. Conversion uses exact multiplicative factors (for example 1 km = 1000 m, 1 hour = 60 minutes). Consistent units permit direct ratio and arithmetic operations. Mixing units without conversion causes errors. Area and volume conversions require squared or cubed conversion factors (1 m^2 = 10,000 cm^2, 1 m^3 = 1,000,000 cm^3).
Compound units and unit rates
Compound units join two measures into a single rate, such as metres per second (m/s) or pounds per kilogram (price per kg). Unit rates express one quantity per one unit of another and simplify comparison. Compound-unit problems require algebraic manipulation of both the numeric ratio and the units. Cancellation of units aids simplification and verification of results.
Scaling, similarity and proportional change
Scaling multiplies all quantities in a ratio by the same factor, preserving proportional relationships. Geometric scaling changes linear measures by a factor, area by the square of the factor and volume by the cube. Proportional reasoning determines unknowns by setting equivalent ratios or using multiplicative factors. Direct proportion uses equality of ratios; inverse proportion uses product equality.
Mixing problems and concentrations
Mixing problems use ratios to combine substances while conserving total quantity and relative concentrations. Concentration problems use ratios such as parts per unit (e.g., g per 100 g or percentage) or volume fractions. Solving mixing problems uses the method of parts, weighted averages or algebraic equations to track amounts of each component and final concentration.
Algebraic use of units
Algebraic expressions include units as multiplicative factors and support symbolic manipulation. Variables represent quantities with units; unit analysis (dimensional analysis) checks that equations are homogeneous. Changing units algebraically requires multiplication by conversion factors expressed as fractions equal to one, allowing substitution without changing the physical quantity.
Key notes
Important points to keep in mind